please need help as soon as possible!

Question

A merry-go-round is to be built in a park. The center of the structure is required to be equidistant from three support columns located at Q(4,−2), R(2,−4), and S(0,2). What are the coordinates for the location of the center of the merry-go-round?

A.) (1, −1)

B.) (−1, −1)

C.) (−1, 1)

D.) (1, 1)

Answer 1

Becky:

Let C (x,y) be the center of the circle.

Since Q, R, S are all suppose to lie on the circle, distances CQ, CR, an CS are all equal (they are all radii so to speak).

Using the distance formula,

d(CS) = √((x-0)² +(y-2)²)

d(CR) = √((x-2)²+(y+4)²)

d(CQ) = √((x-4)²+ (y+2)²)

We solve d(CS) = d(CR) and d(CR) =d(CQ)

1) d(CS) = d(CR)

√(x²+(y-2)²) = √((x-2)²+(y+4)²)

x²+(y-2)² = (x-2)² + (y+4)²

x²+(y²-4y+4) = (x²-4x+4) + (y²+8y+16)

4x-12y = 16

2) d(CR) = d(CQ)

√((x-2)²+(y+4)²) = √((x-4)²+ (y+2)²)

(x-2)²+(y+4)² = (x-4)²+ (y+2)²

x²-4x+4 + y²+8y+16 = x²-8x+16 + y²+4y +4

4x+4y =0

You need to solve the system of equations for x and y

4x+4y = 0

4x-12y = 16

I'll let you figure out the last part. Let me know if you need any other help!

Answer 2

A merry-go-round is to be built in a park. The center of the structure is required to be equidistant from three support columns located at Q(4,−2), R(2,−4), and S(0,2). What are the coordinates for the location of the center of the merry-go-round?

Let (X,Y ) be the coordinates of center

Which is equidistant from three points Q,R and S

So (X-4)² +(Y+2)² = ( X-2)² + (Y+4)² = ( X-0)² + (Y -2)² = D1² =D2² =D3² 

Which give X = -Y

Y= -1 and X= 1

(X, Y) = (1,-1)